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On the homotopy type of the Deligne–Mumford compactification

Johannes Ebert and Jeffrey Giansiracusa

Algebraic & Geometric Topology 8 (2008) 2049–2062

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne–Mumford compactification. We give an integral refinement: the classifying space of the Charney–Lee category actually has the same homotopy type as the moduli stack of stable curves, and the étale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney–Lee category.

Deligne–Mumford compactification, moduli of curves, stack, mapping class group, orbit category
Mathematical Subject Classification 2000
Primary: 32G15
Secondary: 30F60, 14A20, 14D22
Received: 16 July 2008
Accepted: 24 September 2008
Published: 5 November 2008
Johannes Ebert
Mathematisches Institut
der Universität Bonn
Beringstrasse 1
53115 Bonn
Jeffrey Giansiracusa
Mathematical Institute
University of Oxford
24–29 St. Giles’
Oxford, OX1 3LB
United Kingdom